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Thread: Convergence True or False Proof

  1. #1
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    Convergence True or False Proof

    If $\displaystyle a_k \to 0$ as $\displaystyle k \to \infty$, and $\displaystyle \displaystyle\sum_{k=1}^{\infty} a_k$ converges, then $\displaystyle a_k \downarrow 0$ as $\displaystyle k \to \infty$.

    I think this is false and my example is $\displaystyle (-1)^k \frac{1}{k^2}$. What you think?
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  2. #2
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    Quote Originally Posted by ThreeLeftsAndHome View Post
    If $\displaystyle a_k \to 0$ as $\displaystyle k \to \infty$, and $\displaystyle \displaystyle\sum_{k=1}^{\infty} a_k$ converges, then $\displaystyle a_k \downarrow 0$ as $\displaystyle k \to \infty$.
    first information is irrelevant since given the convergence of the series, then $\displaystyle a_k\to0$ as $\displaystyle k\to\infty.$

    oh, i think is misread the problem, wonder what does mean $\displaystyle a_k\downarrow0.$
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  3. #3
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    Quote Originally Posted by Krizalid View Post
    first information is irrelevant since given the convergence of the series, then $\displaystyle a_k\to0$ as $\displaystyle k\to\infty.$

    oh, i think is misread the problem, wonder what does mean $\displaystyle a_k\downarrow0.$
    I think it means that $\displaystyle a_k$ is monotone decreasing.
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  4. #4
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    ah well, but it doesn't say if $\displaystyle a_n\ge0,$ so the counterexample is correct.
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